Conversely if the permutation is a cycle then I think $\varphi$ is "indecomposable" in the sense that it doesn't preserve a nontrivial direct summand. This is equivalent to your condition in the abelian case. In the nonabelian case I don't know that irreducibility is a good condition; e.g. if $n = 1$ and $S$ has trivial outer automorphism group then every automorphism of $G$ is inner and so preserves some cyclic subgroup.

So I'm sure this is classical but if $S$ is nonabelian it looks to me like the automorphism group is the wreath product $\text{Aut}(S) \wr S_n$; basically the point is that the obvious copies of $S$ are uniquely determined by some property (e.g. I think they are the only copies of $S$ which are normal, or which are direct summands) so an automorphism has to permute them. Then a necessary condition for irreducibility is that the corresponding permutation of the copies of $S$ is a cycle.

@Timothy: aha, so you suggest it's not only a metric space but an ultrametric space?

Yes, it's a lovely paper for a number of reasons and that paragraph inspired this question, but I'd like to go into much more detail than this! E.g. he lumps all the algebraic topology proofs together.

@Joel: I see, thanks. I'm not so interested in independent statements, I had in mind open problems where we think the answer is either yes or no. So in this setting the best we can hope for is an open problem that has not yet been proved equivalent to a $\Sigma_1$ statement, e.g. the Collatz conjecture in the link.

Can you give an explicit example of an open problem which is provably not equivalent to a $\Sigma_1$ statement?

And, again: if it's obvious to you that this is a precisely defined category then please tell me the automorphism group of any nontrivial object!

If you're right that rotations aren't allowed (I can't tell either way) then these are not just free-floating abstract polygons but are, say, embedded in the plane and allowed to translate freely. If that's the case the OP should specify this, and then it raises the question of which gluings are allowed, since some gluings may require rotating the pieces. Do we only perform gluings that can be done using translation? There are many details like this that have not been specified!

@Alec: I am asking whether it is an automorphism (this is why I said "into a congruent polygon"); part of the question is I don't understand which objects are being considered to be the same (not just isomorphic). Explicitly (I am ignoring colors throughout), take a $2 \times 1$ rectangle, cut it into two squares, then glue them back into a rectangle but along a different edge. 1) Is this an automorphism (are the source and target considered the same object)? 2) Is this a nontrivial automorphism (is it considered different from the identity)?

I am unable to extract a precise definition of a category out of what you've written so far. Can you describe the automorphism group of any object? Since you have a groupoid it is determined by the automorphism groups of every isomorphism class of object. For example do rotations of a square with all colors the same count as nontrivial automorphisms? If I cut a polygon into two pieces and reassemble it in a different way into a congruent polygon (again let's assume all colors are the same) does that count as a nontrivial automorphism?